Representing vectors
A vector has both size and direction, which is what distinguishes it from an ordinary number. A translation is described as a vector, written as a column: the top number is the movement right, and the bottom number the movement up, with negatives for left and down.
So a translation of (4, −3) written as a column means 4 right and 3 down.
Vectors are written in three ways, all meaning the same thing: as a column, as a bold or underlined letter such as a, or as two points with an arrow above them, such as AB meaning the vector from A to B.
The reverse vector BA is the same length in the opposite direction, so BA = −AB. That single fact does most of the work in the proof questions.
Adding vectors means following one then the other, and column vectors add top to top and bottom to bottom. Subtracting means adding the reverse. Multiplying by a scalar multiplies both parts, so 3a is three times as long in the same direction, and −2a is twice as long in the opposite direction.
The magnitude — the length — of a column vector comes from Pythagoras: a vector of (3, 4) has length √(3² + 4²) = 5.
- Vector
- a quantity with both size and direction, written as a column in which the top number is the movement right and the bottom number the movement up
Geometric arguments and proofsHigher tier
Vector proof questions look unlike anything else on the paper, but they are mechanical once two ideas are in place.
First, a route between two points can be taken any way round the shape, and every route gives the same vector. So to find AC you might go A to O then O to C, writing AC = AO + OC. Going against the arrow makes that part negative.
Second, the parallel test: if one vector is a scalar multiple of another, the two lines are parallel. If they also share a point, the three points lie on a straight line — which is what "prove ABC is a straight line" is asking.
The method is always the same: write the route in letters first, substitute the given vectors, then simplify and factorise to see whether a multiple appears.
Worked example
OACB is a parallelogram with OA = a and OB = b. M is the midpoint of AC. Show that OM = a + ½b.
- Write the route from O to M in letters: OM = OA + AM.
- OA is given as a.
- AC is the side of the parallelogram parallel to OB, so AC = b.
- M is the midpoint of AC, so AM is half of AC, which is ½b.
- Substitute both parts into the route.
Answer: OM = a + ½b, as required. Note the two things that did the work: identifying AC = b from the parallelogram, and halving it for the midpoint. Almost every vector proof turns on those two moves.
Common mistake
Going against an arrow makes the vector negative. Writing AO = a when the diagram gives OA = a reverses the direction and makes every later step wrong — check the direction of each letter pair as you write the route.
Before you practise
- A vector has size and direction; a column vector is right over up, with negatives for left and down.
- BA = −AB — reversing the direction changes the sign.
- Add column vectors top to top and bottom to bottom; a scalar multiplies both parts.
- The magnitude of a column vector comes from Pythagoras.
- Any route between two points gives the same vector, so choose one you have the pieces for.
- Write the route in letters first, then substitute.
- In a parallelogram, opposite sides are equal vectors — that is usually the missing piece.
- A midpoint means halving the vector along that side.
- If one vector is a scalar multiple of another, the lines are parallel; sharing a point makes them a straight line.